Parabolic subgroup - Knowlpedia

Definition

Let $G$ be a connected reductive algebraic group over a field $k$ . A smooth closed subgroup $P\subseteq G$ is parabolic if the quotient $G/P$ is proper over $k$ .

Over an algebraically closed field, this is equivalent to requiring $P$ to contain a Borel subgroup. Every Borel subgroup is therefore parabolic.

Flag variety

The quotient $G/P$ is a flag variety. For $G=GL_n$ , parabolic subgroups are stabilizers of flags of prescribed dimensions.

References

T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, 1998, Chapter 6. DOI.